Colloquia Patavina: What if we lived in a non integer dimensional world?


Martedi' 17 Febbraio 2009 alle ore 16:00 in aula 1A/150 della Torre Archimede la Professoressa Sylvie Paycha dell'Universita' Blaise Pascal terra' una conferenza della serie Colloquia Patavina.

La Commissione Colloquia
M. A. Garuti, M. Pavon, M. Pitteri, F. Rossi

What if we lived in a non integer dimensional world?

Sylvie Paycha (Université Blaise Pascal)

Life would be much easier... at least for theoretical physicists and mathematicians trying to understand what theoretical physicists are doing. A world free of local anomalies... We shall first discuss how in mathematical terms and using the language of pseudodifferential symbols, "non integer dimension" translates to "non integer order symbols".
Now, it turns out that ordinary integration over R^d (resp. discrete summation over Z^d) which is well defined on Schwartz functions, uniquely extends to an R^d (resp. Z^d) translation invariant linear form, the canonical integral (resp. canonical discrete sum) on non integer order symbols. The canonical integral on non integer order symbols has the expected properties; in particular one can perform integration by parts or changes of variable as one would do with ordinary integration. This is what a posteriori provides a justification for many of the physicists' formal but very efficient computations, since they are performing them in non integer dimensions before going back to four dimensions by analytic continuation.
In contrast, ordinary integration over R^d does not extend to an R^d translation invariant linear form on the whole algebra of symbols. Extending ordinary integration over R^d (discrete sum over Z^d) to a linear form on the whole algebra of symbols leads to regularised integrals (resp. regularised discrete sums) which present discrepancies related to anomalies in physics. They are not translation invariant anymore, and do not have the usual properties of integrals and sums.
This means that when back in four dimensions after analytic continuation, physicists have to deal with anomalies. Discrepancies/anomalies that arise when dealing with integer order symbols, are what make theoretical physicists's lives difficult and even more so the lives of mathematicians trying to understand them!

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